In this paper we study the existence of strong solutions and the asymptotic behavior of nonlocal reaction diffusion-type system of equations, called the bidomain equation. The bidomain model was first proposed by Tung [1]}. It is now the generally accepted model of electrical behavior of cardiac tissue [2]}. The heart muscle consists of so-called excitable cells, which react to electrical stimulus by depolarizing. In other words, the potential difference across the cell membrane changes. This depolarization motivates us to consider the tissue consisting of two separate domains: the intracellular and the extracellular. The corresponding electric potentials are denoted with \(u_i\)  and \(u_e\) , and the difference between these potentials is the transmembrane potential, denoted with \(u:=u_i - u_e\) . Since the current, which passes from one domain to the other, must pass through the cell membrane, the assumptions on the conductive properties of the cell membrane, combined with the conservation of current principle, yield the system of equations. It consists of a pair of partial differential equations of reaction-diffusion type, coupled with an ordinary differential equation:
\({\left\lbrace \begin{array}{ll}\frac{\partial u}{\partial t} + f(u,w) + A_i u_i = I_i \text{ in } (0, +\infty ) \times D,\\\frac{\partial u}{\partial t} + f(u,w) - A_e u_e = -I_e \text{ in } (0, +\infty ) \times D,\\\frac{\partial w}{\partial t} + g(u,w) = 0, \text{ in } (0, +\infty ) \times D,\\\sigma _i \nabla u_i \cdot n = \sigma _e \nabla u_e \cdot n = 0, \text{ in } (0, +\infty ) \times \partial D,\\u(0):=u_i(0) - u_e(0) = u_0, w(0) = w_0 \text{ in } D.\end{array}\right.}\) 
